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I think that it is true, but someone with a grammar book should be able to give a much more convincing answer than my personal opinion. By the way, the meaning of this ～に is much more specific than “purpose.” For example, if you say 旅行記を書くために北海道に行く, the person may write the journal during or after the trip, but 旅行記を書きに北海道に行く, the person intends to write the journal during the trip.
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Tsuyoshi ItoAug 31 '11 at 19:25

Just in case, by “I think that it is true,” I mean “I think that it is true that only movement verbs can take [V-stem]に to express a purpose,” although “purpose” is not necessarily the right explanation of this meaning as I stated in my previous comment.
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Tsuyoshi ItoAug 31 '11 at 22:10

1 Answer
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Yes it is true. According my my grammar dictionary, [Verb in 連用形 + に] can only be used with motion verbs such as 行く, 来る, 帰る, 入る, 出る to mean "to go/come/motion-verb (in order to) do something". The "In order to" becomes explicit when の為に is used, indicating a rather important purpose.

Let me try to break down the composition:

Take 遊ぶ as the verb.

遊ぶ in 連用形* yields 遊び (a playing)(Noun)

*any verb or い-adjective can become a noun when placed in 連用形. This is similar to the English concept of "gerund". E.g. talk is a verb. talking is also a verb. But in the sentence "Is my talking distracting you?" talking here is a noun.

に as a particle indicates a point of space/time/reasoning.

(Compared to から which is a half-line with a start point; from a point in space/time/reasoning)

(Compared to まで, which is the other half-line with an end point; terminating at a point in space/time/reasoning)

And let's also take a motion verb:

行く : to go

Then let's try to piece it back together:

遊びに (The point of reasoning of "a playing")(Since 遊び is not a time or space)

遊びに行く (To go to the point of reasoning of "a playing")**(More naturally parsed as "To go play")

** because 行く goes towards whatever is marked by に

Now let's try for your example:

遊びに食べる (To eat, at the point of reasoning of "a playing"). It cannot be naturally parsed because に does not relate the thing it marks with the verb 食べる. Unlike に+motion verb, whatever に marks is directly related to the motion verb.

The act of eating has nothing inherent to do with the playing. On the other hand, moving has an inherent connection to the destination. See sawa's explanation of に in "に and で revisited"